Pythagorean theorem
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Prepared by: JR Brews April 20/2010
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Pythagorean identity for 7D
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Input xInput yCalculated X-productMultiplication table for octonionTest of theorem
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xyz=x x yj
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x11y11z1-232-2345-5476-321Σ1^20
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x21y21z22-311346-6457-752Σ2^20
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x32y31z3-112-2147-7465-563Σ3^22
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x40y41z4-3-2662-3773-15514Σ4^26
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x51y51z5214-4136-63-27725Σ5^22
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x60y61z6117-7124-42-35536Σ6^27
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x70y71z7125-5234-43-16617Σ7^27
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Sum24
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|x|^27|y|^27|z|^224|x|^2|y|^249
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|x|^2|y|^2 - (x.y)^224
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x.y5x.z0y.z0|x x y|^224
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OKOKOrthogonality
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Table from:http://books.google.com/books?id=_PEWt18egGgC&pg=PA235&dq=multiplication+octonion&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=0&cd=6#v=onepage&q=multiplication%20octonion&f=falseLev Vasilʹevitch Sabinin, Larissa Sbitneva, I. P. ShestakovNon-associative algebra and its applicationsPythagorean identity is OK if cell to right is zero0Sum is direct evaluation of |x x y|^2: See reference at right, page 4.http://docs.google.com/viewer?a=v&q=cache:rDnOA-ZKljkJ:www.owlnet.rice.edu/~fjones/chap7.pdf+lagrange%27s+identity+in+the+seven+dimensional+cross+product&hl=en&gl=ph&sig=AHIEtbQQtdVGhgbYhz78SQQb2biLxRi4kA
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